Paths, Cycles, and Flows

One of the key problems in graphs is navigation. In particular, the problem is finding the shortest path between two vertices, whether that is finding the way out of a maze or navigating a road network. The lengths of the shortest paths give rise to a whole collection of natural measures such as the diameter of a graph. If instead of navigating from one vertex to another you would like to traverse the whole graph in some way, you are looking for cycles. Eulerian and Hamiltonian cycles provide paths that traverse every edge or vertex of graph.